Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2

Percentage Accurate: 99.9% → 99.9%

Time: 5.5s

Alternatives: 4

Speedup: 1.2×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x + \left(y \cdot z\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* y z) z)))

C

double code(double x, double y, double z) {
	return x + ((y * z) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * z) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * z) * z);
}

Python

def code(x, y, z):
	return x + ((y * z) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * z) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * z) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y \cdot z\right) \cdot z \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ x (* (* y z) z)))

C

double code(double x, double y, double z) {
	return x + ((y * z) * z);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = x + ((y * z) * z)
end function

Java

public static double code(double x, double y, double z) {
	return x + ((y * z) * z);
}

Python

def code(x, y, z):
	return x + ((y * z) * z)

Julia

function code(x, y, z)
	return Float64(x + Float64(Float64(y * z) * z))
end

MATLAB

function tmp = code(x, y, z)
	tmp = x + ((y * z) * z);
end

Wolfram

code[x_, y_, z_] := N[(x + N[(N[(y * z), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z \cdot y, z, x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (* z y) z x))

C

double code(double x, double y, double z) {
	return fma((z * y), z, x);
}

Julia

function code(x, y, z)
	return fma(Float64(z * y), z, x)
end

Wolfram

code[x_, y_, z_] := N[(N[(z * y), $MachinePrecision] * z + x), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \left(y \cdot z\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(y \cdot z\right) \cdot z} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z} + x \]

   4. lower-fma.f64 99.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z, z, x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z}, z, x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, z, x\right) \]

   7. lower-*.f64 99.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, z, x\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, z, x\right)} \]
5. Add Preprocessing

Alternative 2: 66.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+42}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= z 3.5e+42) (* -1.0 (- x)) (* (* z y) z)))

C

double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.5e+42) {
		tmp = -1.0 * -x;
	} else {
		tmp = (z * y) * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (z <= 3.5d+42) then
        tmp = (-1.0d0) * -x
    else
        tmp = (z * y) * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (z <= 3.5e+42) {
		tmp = -1.0 * -x;
	} else {
		tmp = (z * y) * z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if z <= 3.5e+42:
		tmp = -1.0 * -x
	else:
		tmp = (z * y) * z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (z <= 3.5e+42)
		tmp = Float64(-1.0 * Float64(-x));
	else
		tmp = Float64(Float64(z * y) * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (z <= 3.5e+42)
		tmp = -1.0 * -x;
	else
		tmp = (z * y) * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[z, 3.5e+42], N[(-1.0 * (-x)), $MachinePrecision], N[(N[(z * y), $MachinePrecision] * z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.50000000000000023e42

   1. Initial program 100.0%

      \[x + \left(y \cdot z\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y \cdot z\right) \cdot z} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z} + x \]

      4. lower-fma.f64 100.0

         \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z, z, x\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z}, z, x\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, z, x\right) \]

      7. lower-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, z, x\right) \]

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, z, x\right)} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right)\right)} \]
   6. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

      4. mul-1-neg N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot {z}^{2}}{x}\right)\right)} - 1\right) \cdot \left(-1 \cdot x\right) \]

      5. associate-/l* N/A

         \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{{z}^{2}}{x}}\right)\right) - 1\right) \cdot \left(-1 \cdot x\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{{z}^{2}}{x}} - 1\right) \cdot \left(-1 \cdot x\right) \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{{z}^{2}}{x} - 1\right)} \cdot \left(-1 \cdot x\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{{z}^{2}}{x}} - 1\right) \cdot \left(-1 \cdot x\right) \]

      9. lower-neg.f64 N/A

         \[\leadsto \left(\color{blue}{\left(-y\right)} \cdot \frac{{z}^{2}}{x} - 1\right) \cdot \left(-1 \cdot x\right) \]

      10. lower-/.f64 N/A

          \[\leadsto \left(\left(-y\right) \cdot \color{blue}{\frac{{z}^{2}}{x}} - 1\right) \cdot \left(-1 \cdot x\right) \]

      11. unpow2 N/A

          \[\leadsto \left(\left(-y\right) \cdot \frac{\color{blue}{z \cdot z}}{x} - 1\right) \cdot \left(-1 \cdot x\right) \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(-y\right) \cdot \frac{\color{blue}{z \cdot z}}{x} - 1\right) \cdot \left(-1 \cdot x\right) \]

      13. mul-1-neg N/A

          \[\leadsto \left(\left(-y\right) \cdot \frac{z \cdot z}{x} - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

      14. lower-neg.f64 90.3

          \[\leadsto \left(\left(-y\right) \cdot \frac{z \cdot z}{x} - 1\right) \cdot \color{blue}{\left(-x\right)} \]

   7. Applied rewrites 90.3%

      \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \frac{z \cdot z}{x} - 1\right) \cdot \left(-x\right)} \]

   8. Taylor expanded in x around inf

      \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 65.6%

       \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]

   if 3.50000000000000023e42 < z

   1. Initial program 99.8%

      \[x + \left(y \cdot z\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{y \cdot {z}^{2}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto y \cdot \color{blue}{\left(z \cdot z\right)} \]

      2. associate-*r* N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z} \]

      3. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot z \]

      5. lower-*.f64 90.0

         \[\leadsto \color{blue}{\left(z \cdot y\right)} \cdot z \]

   5. Applied rewrites 90.0%

      \[\leadsto \color{blue}{\left(z \cdot y\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 3.5 \cdot 10^{+42}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z \cdot y\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ -1 \cdot \left(-x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (* -1.0 (- x)))

C

double code(double x, double y, double z) {
	return -1.0 * -x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-1.0d0) * -x
end function

Java

public static double code(double x, double y, double z) {
	return -1.0 * -x;
}

Python

def code(x, y, z):
	return -1.0 * -x

Julia

function code(x, y, z)
	return Float64(-1.0 * Float64(-x))
end

MATLAB

function tmp = code(x, y, z)
	tmp = -1.0 * -x;
end

Wolfram

code[x_, y_, z_] := N[(-1.0 * (-x)), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[x + \left(y \cdot z\right) \cdot z \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + \left(y \cdot z\right) \cdot z} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z + x} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(y \cdot z\right) \cdot z} + x \]

   4. lower-fma.f64 99.9

      \[\leadsto \color{blue}{\mathsf{fma}\left(y \cdot z, z, x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y \cdot z}, z, x\right) \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, z, x\right) \]

   7. lower-*.f64 99.9

      \[\leadsto \mathsf{fma}\left(\color{blue}{z \cdot y}, z, x\right) \]

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z \cdot y, z, x\right)} \]

5. Taylor expanded in x around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right)\right)} \]
6. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot x\right) \cdot \left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right)} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{y \cdot {z}^{2}}{x} - 1\right) \cdot \left(-1 \cdot x\right)} \]

   4. mul-1-neg N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot {z}^{2}}{x}\right)\right)} - 1\right) \cdot \left(-1 \cdot x\right) \]

   5. associate-/l* N/A

      \[\leadsto \left(\left(\mathsf{neg}\left(\color{blue}{y \cdot \frac{{z}^{2}}{x}}\right)\right) - 1\right) \cdot \left(-1 \cdot x\right) \]

   6. distribute-lft-neg-in N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{{z}^{2}}{x}} - 1\right) \cdot \left(-1 \cdot x\right) \]

   7. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{{z}^{2}}{x} - 1\right)} \cdot \left(-1 \cdot x\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \left(\color{blue}{\left(\mathsf{neg}\left(y\right)\right) \cdot \frac{{z}^{2}}{x}} - 1\right) \cdot \left(-1 \cdot x\right) \]

   9. lower-neg.f64 N/A

      \[\leadsto \left(\color{blue}{\left(-y\right)} \cdot \frac{{z}^{2}}{x} - 1\right) \cdot \left(-1 \cdot x\right) \]

   10. lower-/.f64 N/A

       \[\leadsto \left(\left(-y\right) \cdot \color{blue}{\frac{{z}^{2}}{x}} - 1\right) \cdot \left(-1 \cdot x\right) \]

   11. unpow2 N/A

       \[\leadsto \left(\left(-y\right) \cdot \frac{\color{blue}{z \cdot z}}{x} - 1\right) \cdot \left(-1 \cdot x\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \left(\left(-y\right) \cdot \frac{\color{blue}{z \cdot z}}{x} - 1\right) \cdot \left(-1 \cdot x\right) \]

   13. mul-1-neg N/A

       \[\leadsto \left(\left(-y\right) \cdot \frac{z \cdot z}{x} - 1\right) \cdot \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   14. lower-neg.f64 89.5

       \[\leadsto \left(\left(-y\right) \cdot \frac{z \cdot z}{x} - 1\right) \cdot \color{blue}{\left(-x\right)} \]

7. Applied rewrites 89.5%

   \[\leadsto \color{blue}{\left(\left(-y\right) \cdot \frac{z \cdot z}{x} - 1\right) \cdot \left(-x\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]
9. Step-by-step derivation

10. Applied rewrites 55.5%

    \[\leadsto -1 \cdot \left(-\color{blue}{x}\right) \]

11. Final simplification 55.5%

    \[\leadsto -1 \cdot \left(-x\right) \]
12. Add Preprocessing

Alternative 4: 2.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ -x \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (- x))

C

double code(double x, double y, double z) {
	return -x;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = -x
end function

Java

public static double code(double x, double y, double z) {
	return -x;
}

Python

def code(x, y, z):
	return -x

Julia

function code(x, y, z)
	return Float64(-x)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -x;
end

Wolfram

code[x_, y_, z_] := (-x)

Derivation

1. Initial program 99.9%

   \[x + \left(y \cdot z\right) \cdot z \]
2. Add Preprocessing

4. Applied rewrites 11.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left({\left(z \cdot y\right)}^{2}, \frac{z \cdot z}{\left(z \cdot y\right) \cdot z - x}, x \cdot \frac{x}{\left(z \cdot y\right) \cdot z - x}\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{-1 \cdot x} \]
6. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(x\right)} \]

   2. lower-neg.f64 2.3

      \[\leadsto \color{blue}{-x} \]

7. Applied rewrites 2.3%

   \[\leadsto \color{blue}{-x} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024326 
(FPCore (x y z)
  :name "Statistics.Sample:robustSumVarWeighted from math-functions-0.1.5.2"
  :precision binary64
  (+ x (* (* y z) z)))